Stewart Platform - Dynamics Study

1. Compare external forces and forces applied by the actuators
2. Comparison of the static transfer function and the Compliance matrix
- 2.1. Analysis
- 2.2. Conclusion

1 Compare external forces and forces applied by the actuators

1.1 Comparison with fixed support

stewart = initializeStewartPlatform();
stewart = initializeFramesPositions(stewart, 'H', 90e-3, 'MO_B', 45e-3);
stewart = generateGeneralConfiguration(stewart);
stewart = computeJointsPose(stewart);
stewart = initializeStrutDynamics(stewart);
stewart = initializeJointDynamics(stewart, 'type_F', 'universal_p', 'type_M', 'spherical_p');
stewart = initializeCylindricalPlatforms(stewart);
stewart = initializeCylindricalStruts(stewart);
stewart = computeJacobian(stewart);
stewart = initializeStewartPose(stewart);
stewart = initializeInertialSensor(stewart, 'type', 'none');

ground = initializeGround('type', 'none');
payload = initializePayload('type', 'none');

Estimation of the transfer function from \(\bm{\tau}\) to \(\mathcal{\bm{X}}\):

%% Options for Linearized
options = linearizeOptions;
options.SampleTime = 0;

%% Name of the Simulink File
mdl = 'stewart_platform_model';

%% Input/Output definition
clear io; io_i = 1;
io(io_i) = linio([mdl, '/Controller'],              1, 'openinput');  io_i = io_i + 1; % Actuator Force Inputs [N]
io(io_i) = linio([mdl, '/Relative Motion Sensor'],  1, 'openoutput'); io_i = io_i + 1; % Position/Orientation of {B} w.r.t. {A}

%% Run the linearization
G = linearize(mdl, io, options);
G.InputName  = {'F1', 'F2', 'F3', 'F4', 'F5', 'F6'};
G.OutputName = {'Edx', 'Edy', 'Edz', 'Erx', 'Ery', 'Erz'};

Gc = minreal(G*inv(stewart.kinematics.J'));
Gc.InputName = {'Fnx', 'Fny', 'Fnz', 'Mnx', 'Mny', 'Mnz'};

Estimation of the transfer function from \(\bm{\mathcal{F}}_{\text{ext}}\) to \(\mathcal{\bm{X}}\):

%% Input/Output definition
clear io; io_i = 1;
io(io_i) = linio([mdl, '/Disturbances'], 1, 'openinput', [], 'F_ext');  io_i = io_i + 1; % External forces/torques applied on {B}
io(io_i) = linio([mdl, '/Relative Motion Sensor'],  1, 'openoutput'); io_i = io_i + 1; % Position/Orientation of {B} w.r.t. {A}

%% Run the linearization
Gd = linearize(mdl, io, options);
Gd.InputName  = {'Fex', 'Fey', 'Fez', 'Mex', 'Mey', 'Mez'};
Gd.OutputName = {'Edx', 'Edy', 'Edz', 'Erx', 'Ery', 'Erz'};

1.2 Comparison with a flexible support

We redo the identification for when the Stewart platform is on a flexible support.

ground = initializeGround('type', 'flexible');

Estimation of the transfer function from \(\bm{\tau}\) to \(\mathcal{\bm{X}}\):

%% Options for Linearized
options = linearizeOptions;
options.SampleTime = 0;

%% Name of the Simulink File
mdl = 'stewart_platform_model';

%% Input/Output definition
clear io; io_i = 1;
io(io_i) = linio([mdl, '/Controller'],              1, 'openinput');  io_i = io_i + 1; % Actuator Force Inputs [N]
io(io_i) = linio([mdl, '/Relative Motion Sensor'],  1, 'openoutput'); io_i = io_i + 1; % Position/Orientation of {B} w.r.t. {A}

%% Run the linearization
G = linearize(mdl, io, options);
G.InputName  = {'F1', 'F2', 'F3', 'F4', 'F5', 'F6'};
G.OutputName = {'Edx', 'Edy', 'Edz', 'Erx', 'Ery', 'Erz'};

Gc = minreal(G*inv(stewart.kinematics.J'));
Gc.InputName = {'Fnx', 'Fny', 'Fnz', 'Mnx', 'Mny', 'Mnz'};

Estimation of the transfer function from \(\bm{\mathcal{F}}_{\text{ext}}\) to \(\mathcal{\bm{X}}\):

%% Input/Output definition
clear io; io_i = 1;
io(io_i) = linio([mdl, '/Disturbances'], 1, 'openinput', [], 'F_ext');  io_i = io_i + 1; % External forces/torques applied on {B}
io(io_i) = linio([mdl, '/Relative Motion Sensor'],  1, 'openoutput'); io_i = io_i + 1; % Position/Orientation of {B} w.r.t. {A}

%% Run the linearization
Gd = linearize(mdl, io, options);
Gd.InputName  = {'Fex', 'Fey', 'Fez', 'Mex', 'Mey', 'Mez'};
Gd.OutputName = {'Edx', 'Edy', 'Edz', 'Erx', 'Ery', 'Erz'};

1.3 Conclusion

The transfer function from forces/torques applied by the actuators on the payload \(\bm{\mathcal{F}} = \bm{J}^T \bm{\tau}\) to the pose of the mobile platform \(\bm{\mathcal{X}}\) is the same as the transfer function from external forces/torques to \(\bm{\mathcal{X}}\) as long as the Stewart platform’s base is fixed.

2 Comparison of the static transfer function and the Compliance matrix

2.1 Analysis

Initialization of the Stewart platform.

stewart = initializeStewartPlatform();
stewart = initializeFramesPositions(stewart, 'H', 90e-3, 'MO_B', 45e-3);
stewart = generateGeneralConfiguration(stewart);
stewart = computeJointsPose(stewart);
stewart = initializeStrutDynamics(stewart);
stewart = initializeJointDynamics(stewart, 'type_F', 'universal_p', 'type_M', 'spherical_p');
stewart = initializeCylindricalPlatforms(stewart);
stewart = initializeCylindricalStruts(stewart);
stewart = computeJacobian(stewart);
stewart = initializeStewartPose(stewart);
stewart = initializeInertialSensor(stewart, 'type', 'none');

ground = initializeGround('type', 'none');
payload = initializePayload('type', 'none');

Estimation of the transfer function from \(\mathcal{\bm{F}}\) to \(\mathcal{\bm{X}}\):

%% Options for Linearized
options = linearizeOptions;
options.SampleTime = 0;

%% Name of the Simulink File
mdl = 'stewart_platform_model';

%% Input/Output definition
clear io; io_i = 1;
io(io_i) = linio([mdl, '/F'], 1, 'openinput');  io_i = io_i + 1;
io(io_i) = linio([mdl, '/X'], 1, 'openoutput'); io_i = io_i + 1;

%% Input/Output definition
clear io; io_i = 1;
io(io_i) = linio([mdl, '/Controller'],              1, 'openinput');  io_i = io_i + 1; % Actuator Force Inputs [N]
io(io_i) = linio([mdl, '/Relative Motion Sensor'],  1, 'openoutput'); io_i = io_i + 1; % Position/Orientation of {B} w.r.t. {A}

%% Run the linearization
G = linearize(mdl, io, options);
G.InputName  = {'F1', 'F2', 'F3', 'F4', 'F5', 'F6'};
G.OutputName = {'Edx', 'Edy', 'Edz', 'Erx', 'Ery', 'Erz'};

Gc = minreal(G*inv(stewart.kinematics.J'));
Gc.InputName = {'Fnx', 'Fny', 'Fnz', 'Mnx', 'Mny', 'Mnz'};

Let’s first look at the low frequency transfer function matrix from \(\mathcal{\bm{F}}\) to \(\mathcal{\bm{X}}\).

4.7e-08	-7.2e-19	5.0e-18	-8.9e-18	3.2e-07	9.9e-18
4.7e-18	4.7e-08	-5.7e-18	-3.2e-07	-1.6e-17	-1.7e-17
3.3e-18	-6.3e-18	2.1e-08	4.4e-17	6.6e-18	7.4e-18
-3.2e-17	-3.2e-07	6.2e-18	5.2e-06	-3.5e-16	6.3e-17
3.2e-07	2.7e-17	4.8e-17	-4.5e-16	5.2e-06	-1.2e-19
4.0e-17	-9.5e-17	8.4e-18	4.3e-16	5.8e-16	1.7e-06

And now at the Compliance matrix.

4.7e-08	-2.0e-24	7.4e-25	5.9e-23	3.2e-07	5.9e-24
-7.1e-25	4.7e-08	2.9e-25	-3.2e-07	-5.4e-24	-3.3e-23
7.9e-26	-6.4e-25	2.1e-08	1.9e-23	5.3e-25	-6.5e-40
1.4e-23	-3.2e-07	1.3e-23	5.2e-06	4.9e-22	-3.8e-24
3.2e-07	7.6e-24	1.2e-23	6.9e-22	5.2e-06	-2.6e-22
7.3e-24	-3.2e-23	-1.6e-39	9.9e-23	-3.3e-22	1.7e-06

2.2 Conclusion

The low frequency transfer function matrix from \(\mathcal{\bm{F}}\) to \(\mathcal{\bm{X}}\) corresponds to the compliance matrix of the Stewart platform.